skip to main content

Title: Surface wave pattern formation in a cylindrical container
Surface waves are excited by mechanical vibration of a cylindrical container having an air/water interface pinned at the rim, and the dynamics of pattern formation is analysed from both an experimental and theoretical perspective. The wave conforms to the geometry of the container and its spatial structure is described by the mode number pair ( $n,\ell$ ) that is identified by long exposure time white light imaging. A laser light system is used to detect the surface wave frequency, which exhibits either a (i) harmonic response for low driving amplitude edge waves or (ii) sub-harmonic response for driving amplitude above the Faraday wave threshold. The first 50 resonant modes are discovered. Control of the meniscus geometry is used to great effect. Specifically, when flat, edge waves are suppressed and only Faraday waves are observed. For a concave meniscus, edge waves are observed and, at higher amplitudes, Faraday waves appear as well, leading to complicated mode mixing. Theoretical predictions for the natural frequency of surface oscillations for an inviscid liquid in a cylindrical container with a pinned contact line are made using the Rayleigh–Ritz procedure and are in excellent agreement with experimental results.
Authors:
; ; ;
Award ID(s):
1935590
Publication Date:
NSF-PAR ID:
10311615
Journal Name:
Journal of Fluid Mechanics
Volume:
915
ISSN:
0022-1120
Sponsoring Org:
National Science Foundation
More Like this
  1. Surface waves are excited at the boundary of a mechanically vibrated cylindrical container and are referred to as edge waves. Resonant waves are considered, which are formed by a travelling wave formed at the edge and constructively interfering with its centre reflection. These waves exhibit an axisymmetric spatial structure defined by the mode number $n$ . Viscoelastic effects are investigated using two materials with tunable properties; (i) glycerol/water mixtures (viscosity) and (ii) agarose gels (elasticity). Long-exposure white-light imaging is used to quantify the magnitude of the wave slope from which frequency-response diagrams are obtained via frequency sweeps. Resonance peaks and bandwidths are identified. These results show that for a given $n$ , the resonance frequency decreases with viscosity and increases with elasticity. The amplitude of the resonance peaks are much lower for gels and decrease further with mode number, indicating that much larger driving amplitudes are needed to overcome the elasticity and excite edge waves. The natural frequencies for a viscoelastic fluid in a cylindrical container with a pinned contact-line are computed from a theoretical model that depends upon the dimensionless Ohnesorge number ${Oh}$ , elastocapillary number ${Ec}$ and Bond number ${Bo}$ . All show good agreement with experimental observations.more »The eigenvalue problem is equivalent to the classic damped-driven oscillator model on linear operators with viscosity appearing as a damping force and elasticity and surface tension as restorative forces, consistent with our physical interpretation of these viscoelastic effects.« less
  2. Recent experiments have observed the emergence of standing waves at the free surface of elastic bodies attached to a rigid oscillating substrate and subjected to critical values of forcing frequency and amplitude. This phenomenon, known as Faraday instability, is now well understood for viscous fluids but surprisingly eluded any theoretical explanation for soft solids. Here, we characterize Faraday waves in soft incompressible slabs using the Floquet theory to study the onset of harmonic and subharmonic resonance eigenmodes. We consider a ground state corresponding to a finite homogeneous deformation of the elastic slab. We transform the incremental boundary value problem into an algebraic eigenvalue problem characterized by the three dimensionless parameters, that characterize the interplay of gravity, capillary and elastic waves. Remarkably, we found that Faraday instability in soft solids is characterized by a harmonic resonance in the physical range of the material parameters. This seminal result is in contrast to the subharmonic resonance that is known to characterize viscous fluids, and opens the path for using Faraday waves for a precise and robust experimental method that is able to distinguish solid-like from fluid-like responses of soft matter at different scales.
  3. Abstract Englacial water transport is an integral part of the glacial hydrologic system, yet the geometry of englacial structures remains largely unknown. In this study, we explore the excitation of fluid resonance by small amplitude waves as a probe of englacial geometry. We model a hydraulic network consisting of one or more tabular cracks that intersect a cylindrical conduit, subject to oscillatory wave motion initiated at the water surface. Resulting resonant frequencies and quality factors are diagnostic of fluid properties and geometry of the englacial system. For a single crack–conduit system, the fundamental mode involves gravity-driven fluid sloshing between the conduit and the crack, at frequencies between 0.02 and 10 Hz for typical glacial parameters. Higher frequency modes include dispersive Krauklis waves generated within the crack and tube waves in the conduit. But we find that crack lengths are often well constrained by fundamental mode frequency and damping rate alone for settings that include alpine glaciers and ice sheets. Branching crack geometry and dip, ice thickness and source excitation function help define limits of crack detectability for this mode. In general, we suggest that identification of eigenmodes associated with wave motion in time series data may provide a pathway towardmore »inferring englacial hydrologic structures.« less
  4. This paper presents a theoretical and experimental study of the long-standing fluid mechanics problem involving the temporal resolution of a large localised initial disturbance into a sequence of solitary waves. This problem is of fundamental importance in a range of applications, including tsunami and internal ocean wave modelling. This study is performed in the context of the viscous fluid conduit system – the driven, cylindrical, free interface between two miscible Stokes fluids with high viscosity contrast. Owing to buoyancy-induced nonlinear self-steepening balanced by stress-induced interfacial dispersion, the disturbance evolves into a slowly modulated wavetrain and further into a sequence of solitary waves. An extension of Whitham modulation theory, termed the solitary wave resolution method, is used to resolve the fission of an initial disturbance into solitary waves. The developed theory predicts the relationship between the initial disturbance’s profile, the number of emergent solitary waves and their amplitude distribution, quantifying an extension of the well-known soliton resolution conjecture from integrable systems to non-integrable systems that often provide a more accurate modelling of physical systems. The theoretical predictions for the fluid conduit system are confirmed both numerically and experimentally. The number of observed solitary waves is consistently within one to two wavesmore »of the prediction, and the amplitude distribution shows remarkable agreement. Universal properties of solitary wave fission in other fluid dynamics problems are identified.« less
  5. Pile driving is used for constructing foundation supports for offshore structures. Underwater noise, induced by in-water pile driving, could adversely impact marine life near the piling location. Many studies have computed this noise in close ranges by using semi-analytical models and Finite Element Method (FEM) models. This work presents a Spectral Element Method (SEM) wave simulator as an alternative simulation tool to obtain close-range underwater piling noise in complex, fully three-dimensional, axially-asymmetric settings in the time domain for impacting force signals with high-frequency contents (e.g., frequencies greater than 1000[Formula: see text]Hz). The presented numerical results show that the flexibility of SEM can accommodate the axially-asymmetric geometry of a model, its heterogeneity, and fluid-solid coupling. We showed that there are multiple Mach Cones of different angles in fluid and sediment caused by the difference in wave speeds in fluid, a pile, and sediment. The angles of Mach Cones in our numerical results match those that are theoretically evaluated. A previous work 18 had shown that Mach Cone waves lead to intense amplitudes of underwater piling noise via a FEM simulation in an axis-symmetric setting. Since it modeled sediment as fluid with a larger wave speed than that of water, we examinedmore »if our SEM simulation, using solid sediment–fluid coupling, leads to additional Mach Cones. Because this work computes the shear wave in sediment and the downward-propagating shear wave in a pile, we present six Mach Cones in fluid and sediment induced by downward-propagating P- and S-waves in a pile in lieu of two previously-reported Mach Cones in fluid and sediment (modeled as fluid) induced by a downward-propagating P-wave in a pile. We also showed that the amplitudes of the close-range underwater noise are dependent on the cross-sectional geometry of a pile. In addition, when a pile is surrounded by a solid of an axially-asymmetric geometry, waves are reflected from the surface of the surrounding solid back to the fluid so that constructive and destructive interferences of waves take place in the fluid and affect the amplitude of the underwater piling noise.« less